As the variable is replaced with in the integral
each value of in the integral needs to be associated with the appropriate value of the field which is .The new field represents a remapping of the original field .Each value in the new field with coordinates corresponds to the value in the field with coordinates .
The remapping can be understood easier if one takes the particular case ky=0. The change of variable in (5) becomes
and the zero-offset migrated field becomes
by a new variable to get back the inverse Fourier transform of the initial expression.
The integration in kh represents the inverse Fourier transform at zero offset (h=0). However because equation (12) actually performs summation along hyperbolas in the spatial coordinate h, replacing the integral in kh by the inverse Fourier transform will not return the constant-offset sections.
Applying the same technique to the case I obtain